Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, p\neq 0$. $\dfrac{{(a^{-1}p^{5})^{-1}}}{{(a^{4}p^{-5})^{2}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{-1}p^{5})^{-1} = (a^{-1})^{-1}(p^{5})^{-1}}$ On the left, we have ${a^{-1}}$ to the exponent ${-1}$ . Now ${-1 \times -1 = 1}$ , so ${(a^{-1})^{-1} = a}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{-1}p^{5})^{-1}}}{{(a^{4}p^{-5})^{2}}} = \dfrac{{ap^{-5}}}{{a^{8}p^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{ap^{-5}}}{{a^{8}p^{-10}}} = \dfrac{{a}}{{a^{8}}} \cdot \dfrac{{p^{-5}}}{{p^{-10}}} = a^{{1} - {8}} \cdot p^{{-5} - {(-10)}} = a^{-7}p^{5}$